Unstable manifolds of a class of delayed partial differential equations with nondense domain

Chiraz Jendoubi¹

¹ Faculty of Sciences University of Sfax Sfax, Tunisia

Annales Polonici Mathematici, Tome 118 (2016) no. 2-3, pp. 181-208

Résumé

We present an unstable manifold theory for the abstract delayed semilinear Cauchy problem with nondense domain $$ \frac {du}{dt}=(A+B(t))u(t)+f(t,u_t),\hskip 1em t\in \mathbb {R}, $$ where $(A,D(A))$ satisfies the Hille–Yosida condition, $(B(t))_{t\in \mathbb {R}}$ is a family of operators in $\mathcal {L}(\overline {D(A)},X)$ satisfying some measurability and boundedness conditions, and the nonlinear forcing term $f$ satisfies $\| f(t,\phi )-f(t,\psi )\| \leq \varphi (t)\| \phi -\psi \| _{\mathcal {C}}$. Here $\varphi $ belongs to some admissible spaces and $\phi , \psi \in \mathcal {C}:=C([-r,0],X)$. To reach our goal, we rely mainly on extrapolation theory. First, we develop a new variation of constants formula adapted to our problem. Then, using the characterization of exponential dichotomy, the properties of admissible spaces, the Lyapunov–Perron method as well as useful technical structures we prove the existence of an unstable manifold for our solutions. We also state an exponential attractiveness result concerning the unstable manifold. For illustration, we give an example.

Détail

DOI : 10.4064/ap3913-11-2016

Keywords: present unstable manifold theory abstract delayed semilinear cauchy problem nondense domain frac t hskip mathbb where satisfies hille yosida condition mathbb family operators mathcal overline satisfying measurability boundedness conditions nonlinear forcing term satisfies phi f psi leq varphi phi psi mathcal here varphi belongs admissible spaces phi psi mathcal r reach rely mainly extrapolation theory first develop variation constants formula adapted problem using characterization exponential dichotomy properties admissible spaces lyapunov perron method useful technical structures prove existence unstable manifold solutions state exponential attractiveness result concerning unstable manifold illustration example